(JKOMKTKICAL 



f TIVI-:. 



vards, a repetition of tho process, when, in a future 



. iirios in size and form from the ono below, the 



<lit!iculti"< will not be so great. Probably it will bo advisable to 



nhito H..III.I of the work, to prevent failure. Hake a b 



< .;u:il :.. tho diHtunoe tho object in to tho left of the eye; draw 



tuuku 6 /equal to tho distance 

 tin- in are.st auglo in within tin- ]>i<-ture; 

 iin.l because tho line from b vanishes 

 at PS, thoroforo tho lino from /, to out 

 it. within, must bo drawn to 

 m: 1 , tho distance point of K or P8, to 

 .li-t.-niiiM- the nearest part of the ob- 

 (Somo writers on Perspective 

 all tli.' DK tho DPS, meaning the dis- 

 of the point of sight. It makes 

 no i>r:u-tii-:il ( 1 i nVrenop, because DE I or 

 DE a represents tho space between tho 

 oyo and tho picture plane, that is, be- 

 tween x and PH ; rs being on the pic- 

 ture plane, which is supposed to be in 

 a perpendicular position ; the line 

 below, marked PP, being its base. See 

 Fig. 21, Vol. II., page 360.) Through 

 <, directed from DVP 1 , draw a line to 

 r ; make r s equal to the length of the 

 block ; draw from 

 s back again to 

 DVP 1 , which will 

 cut the vanishing 

 line from c to vp 1 

 in tn; cm will 

 then be the per- 

 spective represen- 

 tation of tin- 

 length of the 

 block. Through 

 c, directed from 

 DVP*, draw a line 

 to n; make n o 

 equal to tho width 

 of the block, and 

 rule from o back 

 again to DVP* ; 

 this will cut the 

 line from c to vp 2 

 in v; c v will be 

 the width of the 

 block. We trust 

 the remainder of 

 the work, includ- 

 ing the thickness 

 of tho block, will 

 present no difficul- 

 ties. 



We will make 

 further use of this 

 problem, by chang- 

 ing tho proportions 

 of tho upper block 

 to 4 feet long, and 

 2 feet wide ; its 

 plan being in the 

 centre of the plan 

 of the lower one. 

 In this case a plan 

 must be drawn 

 both of tho blocks 

 and the PP, to show 

 how tho former are 

 situated and con- 

 nected with the latter, and from which we obtain the propor- 

 tions and distanceo of tho several parts from ono another 

 and from the PP. Therefore Fig. 58 is the first considera- 

 tion ; it is a plan constructed according to tho particulars 

 given in the question. Draw the PP. Anywhere, say from 

 o, draw tho line a I, at an angle of 40 with tho PP. Upon 

 this last line find the point d, the nearest angle within the 

 Pr; draw d c perpendicularly with tho PP; place PS 3 feet 



Fig. 60. 



to the right of c d. Upon d draw UM plan of th* lower 

 block ; afterward! the plan of the upper one, e k i kt ail iU 

 idea being one foot within the larger plan. la Fig. 59 we 

 haw e represented only UM upper block ; UM lower OM will be 

 imply a repetition of the one in Fig. 57, which oar pupils 

 most not omit repeating when drawing 

 Fig. 59. We will raw eommeiM* with 

 the n L, and proceed upward*. Tbe 

 n, DB, PP, and n will b* UM same 

 as in Fig. 57. The dutanoe of UM 

 nearest angle a from the tr must be 

 measured from b to e on UM vr, and 

 equal to n e, taken from Fig 

 The distance ef, of the point a within, 

 mast be equal to/ e (Fig. 58). Draw 

 from a to VP', and also the other way 

 to the PP in m j a line from n perpen- 

 dicularly to PP will be the line of eon- 

 tact, upon which to measure the thick- 

 ness, m n, of the block. Tho length 

 and breadth to bo cut off on the line* 

 which vanish to VP 1 and VP* mutt be 

 taken from the plan, viz., e K for 

 the length, and e k for the breadth, 

 as shown in e h and o k (Fig. 59). It 

 will be noticed 

 that the difference 

 of dimensions be- 

 tween the two 

 blocks, and the 

 greater distance 

 of tho lower block 

 from the PP, causes 

 a change of posi- 

 tion for the line of 

 contact, or rather, 

 another line of 

 contact most be in- 

 troduced. Tho per- 

 pendicular from i 

 is the line of con- 

 tact for the lower 

 block, whilst the 

 one from m will be 

 the line of contact 

 for the upper; 

 proving that in all 

 cases the first part 

 of the construction 

 to be considered ia 

 the position of the 

 nearest point of 

 the object, with 

 regard to the eye 

 and the PP ; leav- 

 ing the rest to 

 whatever may re- 

 sult from the work, 

 according to the 

 varied character 

 of tho subject, and 

 the conditions 

 given in the state- 

 ment. 



Before we mako 

 any further appli- 

 cation of tho rule 

 and process of the 

 above problem, wo 

 will explain ano- 

 ther important step connected with this part of our subject, 

 and afterwards combine the two in an especial case. 



Our next consideration will bo tho way in which we can make 

 use of a dinyonal line for determining retiring distances and re- 

 tiring proportions ; that is, tho angle which the diagonal makes 

 with tho PP (we will suppose it to be the diagonal of a square). 

 The diagonal is obtained by bisecting tho angle formed by the 

 vanishing lines from to VP 1 and VP : its VP and distance point 



